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The Cauchy problem and the integral operator

Equation () describes a continuous process of reflected wavefield continuation in the time-offset-midpoint domain. In order to find an integral-type operator that performs the one-step offset continuation, I consider the following initial-value (Cauchy) problem for equation ():

Given a post-NMO constant-offset section at half-offset h1
 (219)
and its first-order derivative with respect to offset
 (220)
find the corresponding section P(0)(tn,y) at offset h.

Equation () belongs to the hyperbolic type, with the offset coordinate h being a time-like'' variable and the midpoint coordinate y and the time tn being space-like'' variables. The last condition () is required for the initial value problem to be well-posed Courant (1962). From a physical point of view, its role is to separate the two different wave-like processes embedded in equation (), which are analogous to inward and outward wave propagation. We will associate the first process with continuation to a larger offset and the second one with continuation to a smaller offset. Though the offset derivatives of data are not measured in practice, they can be estimated from the data at neighboring offsets by a finite-difference approximation. Selecting a propagation branch explicitly, for example by considering the high-frequency asymptotics of the continuation operators, can allow us to eliminate the need for condition (). In this section, I discuss the exact integral solution of the OC equation and analyze its asymptotics.

The integral solution of problem (-) for equation () is obtained in Appendix  with the help of the classic methods of mathematical physics. It takes the explicit form

 (221)
where the Green's functions G0 and G1 are expressed as
 (222) (223)
and the parameter is
 (224)
H stands for the Heavyside step-function.

From equations () and () one can see that the impulse response of the offset continuation operator is discontinuous in the time-offset-midpoint space on a surface defined by the equality
 (225)
which describes the wavefronts'' of the offset continuation process. In terms of the theory of characteristics Courant (1962), the surface corresponds to the characteristic conoid formed by the bi-characteristics of equation () - time rays emerging from the point . The common-offset slices of the characteristic conoid are shown in the left plot of Figure .

con
Figure 7
Constant-offset sections of the characteristic conoid - offset continuation fronts'' (left), and branches of the conoid used in the integral OC operator (right). The upper part of the plots (small times) corresponds to continuation to smaller offsets; the lower part (large times) corresponds to larger offsets.

As a second-order differential equation of the hyperbolic type, equation () describes two different processes. The first process is forward'' continuation from smaller to larger offsets, the second one is reverse'' continuation in the opposite direction. These two processes are clearly separated in the high-frequency asymptotics of operator (). To obtain the asymptotical representation, it is sufficient to note that is the impulse response of the causal half-order integration operator and that is asymptotically equivalent to (t, a >0). Thus, the asymptotical form of the integral offset-continuation operator becomes

 (226)
Here the signs +'' and -'' correspond to the type of continuation (the sign of h-h1), and stand for the operators of causal and anticausal half-order differentiation and integration applied with respect to the time variable tn, the summation paths correspond to the two non-negative sections of the characteristic conoid () (Figure ):

 (227)
where , and ; is the midpoint separation (the integration parameter), and and are the following weighting functions:

 (228) (229)
Expression () for the summation path of the OC operator was obtained previously by Stovas and Fomel (1993, 1996) and Biondi and Chemingui (1994a,b). A somewhat different form of it is proposed by Bagaini and Spagnolini (1996). I describe the kinematic interpretation of formula () in Appendix .

In the high-frequency asymptotics, it is possible to replace the two terms in equation () with a single term Fomel (1996b). The single-term expression is
 (230)
where
 (231) (232)
A more general approach to true-amplitude asymptotic offset continuation is developed by Santos et al. (1997).

The limit of expression () for the output offset h approaching zero can be evaluated by L'Hospitale's rule. As one would expect, it coincides with the well-known expression for the summation path of the integral DMO operator Deregowski and Rocca (1981)
 (233)
I discuss the connection between offset continuation and DMO in the next section.

Next: Offset continuation and DMO Up: Offset continuation for reflection Previous: Kirchhoff model and the
Stanford Exploration Project
12/28/2000